Power Concavity for Solutions of Nonlinear Elliptic Problems in Convex Domains
We investigate convexity properties of solutions to elliptic Dirichlet problems in convex domains. In particular we give conditions on the operator F such that a suitable power of a positive solution u of a fully nonlinear equation F(x,u,Du,D 2 u)=0 in a
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Abstract We investigate convexity properties of solutions to elliptic Dirichlet problems in convex domains. In particular we give conditions on the operator F such that a suitable power of a positive solution u of a fully nonlinear equation F (x, u, Du, D 2 u) = 0 in a convex domain Ω, vanishing on ∂Ω, is concave. Keywords Convexity of solutions · Power concavity · Elliptic equations
1 Introduction Let Ω ⊂ Rn be a convex set. We will deal with problems of the following type ⎧ 2 ⎪ ⎨ F (x, u, Du, D u) = 0 in Ω, (1) u=0 on ∂Ω, ⎪ ⎩ u>0 in Ω, where F (x, t, ξ, A) is a real elliptic operator acting on Rn × R × Rn × Sn . Here Du and D 2 u are the gradient and the Hessian matrix of the function u respectively, and Sn is the set of the n × n real symmetric matrices, Ω is a bounded open convex subset of Rn . The goal is to give assumptions that yield the concavity of some power of u. More precisely, we look for p-concave solutions of (1) for some p ≤ 1; we recall hereafter the definition of p-concavity. Definition 1 A positive function u (defined in a convex set) is said p-concave for p p some p = 0 if |p| u is concave, while it is said log-concave (or 0-concave) if log(u) is concave. M. Bianchini · P. Salani (B) Dipartimento di Matematica “U. Dini”, Università di Firenze, viale Morgagni 67/A, 50134 Firenze, Italy e-mail: [email protected] M. Bianchini e-mail: [email protected] R. Magnanini et al. (eds.), Geometric Properties for Parabolic and Elliptic PDE’s, Springer INdAM Series 2, DOI 10.1007/978-88-470-2841-8_3, © Springer-Verlag Italia 2013
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See also the equivalent Definition 2. The case p = 1 corresponds to usual concavity. Many authors investigated this question. It is for instance well known that the solution of the torsional rigidity problem (i.e. F = Δu + 1 in our notation) in a convex domain is 1/2-concave [7] and that any positive eigenfunction of the Laplacian associated to the first Dirichlet eigenvalue in a convex domain is log-concave [3]; see also [7, 15–17] for instance. Refer to [13] for a good presentation of related problems and for a comprehensive bibliography of classical results, more recent related results (and references) are for instance in [1, 2, 5, 6, 18–21, 23, 25, 26]. In the fundamental paper [1] the authors give conditions which ensure the concavity of solutions u of (1). Recently, the method of [1] has been refined in [26], obtaining also some new rearrangement inequalities for solutions of (1) and BrunnMinkowski inequalities for possibly related functionals. Thanks to [1], one can then investigate the p-concavity of u by writing the equation that governs up and proving that this equation satisfies the condition given therein. On the other hand, to choose a suitable p, to write the equation for up and then to verify the assumptions of [1, 26] may be sometimes hard and by no means trivial. In the present paper we show a way to track down a suitable p and retrieve the p-concavity of u directly from (1), without writing the equa
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